Production of hydrocarbon from a well is normally conducted with a constant production rate over long periods, although the rate can be changed during the productive life of the well due to maintenance and other technical requirements. FIG. 1 is a graph illustrating reservoir pressure change with time for a well producing at constant rate in a closed reservoir. Pi is a reservoir initial pressure; Pw is a wellbore pressure; and Pcri is the lowest permissible wellbore pressure (critical pressure). The time sequence of the graph of FIG. 1 is t1<t2<t3<t4<t5<t6 . . . . At the start of a production, reservoir pressure initially depletes in the immediate neighborhood of the wellbore, and this pressure drawdown spreads outward diffusively towards the reservoir outer boundary (as shown in FIG. 1). For a closed (sealed) reservoir, the no-flow reservoir boundary starts to affect the pressure when the spreading pressure depletion front approaches the boundary. When the boundary effect has been fully reflected in the pressure field, the spatial distribution of the pressure no longer changes with time and the fluid flow reaches the so-called pseudo-steady state (lines 102A in FIG. 1). The flow prior to the pseudo-steady state flow is called the transient flow (lines 102B in FIG. 1), the duration of which depends on how fast the pressure drawdown diffuses in the reservoir, which in turn is determined by the reservoir and fluid properties, namely permeability, porosity, viscosity and compressibility. For conventional reservoirs where the permeability is greater than 0.1 mD (mini-Darcy), the transient flow period usually lasts from days to months; while for unconventional reservoirs which have permeabilities less than 0.1 mD, the period can last from years to even tens of years. For closed reservoirs, the pseudo-steady state flow is a dominant, long-duration and most productive flow regime, especially for conventional reservoirs. During the pseudo-steady state flow period, the wellbore bottom-hole flowing pressure (BHFP) decreases linearly in time in order to maintain the constant production rate. However, once the bottom-hole flowing pressure has declined to the lowest permissible value, which is often determined by the surface equipment limitations, a constant rate production can no longer be continued, and a constant pressure production must follow. The production rate for this constant pressure production period declines in time, eventually approaching zero as the reservoir pressure approaches the lowest permissible wellbore pressure (lines 102C in FIG. 1).
Pseudo-steady state flow is a dominant flow regime during constant rate production from a finite, closed reservoir. For a vertically fractured-well in a finite reservoir approximated as having an slightly elliptical shape, conventional solutions exist for analytically determining the flow for the case of infinite fracture conductivity. For finite fracture conductivity, conventional computational techniques to achieve a pseudo-steady state solution involve running numerical simulations over long times of hours, days, or longer.
Pseudo-steady state flow is a dominant flow regime during constant rate production from a closed reservoir: after the effects of the no-flow condition on the reservoir outer boundary have been fully reflected in the flow field and the transients associated with the flow startup have decayed to be negligible, the flow in the reservoir reaches a state in which the spatial distribution of the pressure no longer changes with time. Pseudo-steady state flow is thus a boundary-dominated flow. One definition for pseudo-steady state is the condition in a finite, closed reservoir when producing at a constant rate that “every point within the reservoir will eventually experience a constant rate of pressure decline.” This constant rate of pressure decline is the result of mass conservation for constant rate production from a closed reservoir. This condition is sometimes referred to as pseudo-steady, quasi-steady, semi-steady, or even steady state. The term pseudo-steady is used here in reference to this particular flow regime.
Pseudo-steady state (PSS) can be a prolonged period of constant rate production from a closed reservoir. During this period, the reservoir pressure declines linearly with time, the rate of which is determined by the specified production rate and the drainage area. The pseudo-steady state solution provides the reservoir pressure distribution as well as the productivity index for this important flow period. Once the bottom hole flowing pressure has declined to the lowest permissible value, however, a constant rate production can no longer be continued, and a constant pressure production must follow. The production rate for this latter constant pressure production period declines in time. Production rate decline analysis for this period plays an important role for estimating the hydrocarbon reserves in place and for assessing the economically recoverable amount of fluid from a reservoir. Because pseudo-steady state is the flow regime immediate preceding the production rate decline period, the pseudo-steady state solution has been conventionally used in the production rate decline analysis for unfractured wells and for fractured wells. In these analyses, the pseudo-steady state dimensionless pressure drawdown at the wellbore is expressed asΔpwD,PSS=2πtDA+bD,PSS,  (1)where tDA is the drainage area based dimensionless time, and bD,PSS is the so-called pseudo-steady state constant which depends on the reservoir model as well as the well/reservoir configuration. This pseudo-steady state constant bD,PSS is used to define the appropriate dimensionless decline rate and time in many of the currently used production decline rate analysis models. Furthermore, the pseudo-steady state constant is the reciprocal of the dimensionless productivity index JD,PSS for the pseudo-steady state, JD,PSS=1/bD,PSS, which measures the productivity of the well for this flow period. JD,PSS is also important for production optimization for a fractured well. For unfractured wells, the pseudo-steady state constant bD,PSS can be obtained analytically for reservoirs of very simple shapes. These exact analytical solutions have been modified by shape factors and used as approximate analytical solutions for other reservoir geometries. For hydraulically fractured wells, however, exact analytical solution for bD,PSS is not available. For a vertically fractured well with infinite fracture conductivity, an exact analytical solution for the pseudo-steady state flow in a reservoir bounded by an elliptical boundary is known, which leads to an analytical expression for the pseudo-steady state constant bD,PSS. For the more practical case of finite fracture conductivity, however, no exact analytical solution in the physical variable space has been reported in the literature for pseudo-steady state flow. For finite fracture conductivity, one conventional numerical procedure is to extract bD,PSS by subtracting 2πtDA from the long-time numerical solution for constant rate production from a fractured well in an elliptical reservoir. This procedure is quite time consuming; and curve-fitting has been used to obtain an empirical relation between bD,PSS and the reservoir geometric parameter and the fracture conductivity.